In his book Introduction to the Differential Equations of Physics, German physicist Ludwig Hopf opens with the following statement:

Any differential equation expresses a relation between derivatives or between derivatives and given functions of the variables.  It thus establishes a relation between the increments of certain quantities and these quantities themselves.  This property of a differential equation makes it the natural expression of the principle of causality which is the foundation of exact natural science.  the ancient Greeks established laws of nature in which certain relation between numbers (harmony of spheres) or certain shapes of bodies played a privileged role.  The law was supposed to state something about a process as a whole, or about the complete shape of a body.  In more recent times (Galileo, Newton, etc.) a different concept has been adopted.  We do not try to establish a relation between all phases of a process immediately, but only between one phase and the next.  A law of this type may express, for example, how a certain state will develop in the immediate future, or it may describe the influence of the state of a certain particle on the particles in the immediate neighbourhood.  Thus we have a procedure for the description of a law of nature in terms of small (mathematically speaking, infinitesimal) differences of time and space.  The increments with which the law is concerned appear as derivatives, i.e., as the limits of the quotient of the increments of the variables which describe the process over the increment of space or time in wihch this development takes place.  A law of nature of this form is the expression of the relation between one state and the neighbouring (in time or space) states and therefore represents a special form of the principle of causality.

The whole issue of causality is an important one for both scientific and theological reasons, and I want to touch on one of each.

Every event that takes place in the universe is a result of an event before it.  Those events in turn are the results of those which have gone before.  All of these events form a chain which leads back to the first cause.  The need for the first cause is one of St. Thomas Aquinas’ proofs of God’s existence:

The second way is from the nature of the efficient cause. In the world of sense we find there is an order of efficient causes. There is no case known (neither is it, indeed, possible) in which a thing is found to be the efficient cause of itself; for so it would be prior to itself, which is impossible. Now in efficient causes it is not possible to go on to infinity, because in all efficient causes following in order, the first is the cause of the intermediate cause, and the intermediate is the cause of the ultimate cause, whether the intermediate cause be several, or only one. Now to take away the cause is to take away the effect. Therefore, if there be no first cause among efficient causes, there will be no ultimate, nor any intermediate cause. But if in efficient causes it is possible to go on to infinity, there will be no first efficient cause, neither will there be an ultimate effect, nor any intermediate efficient causes; all of which is plainly false. Therefore it is necessary to admit a first efficient cause, to which everyone gives the name of God.

Although, as Hopf points out, our understanding of how that causality actually works in the physical universe is different from the Greeks (and Thomas Aquinas worked in a Greek concept of natural philosophy) the truth of the importance of causality is undiminished.

To determine what comes after is a major reason for differential equations, which contain three elements: the equation itself, the initial conditions and the boundary conditions.  Once we have these, we can predict the behaviour of a system.  In some cases we can do so with a “simple” equation, others require discretisation and numerical modelling.  And that leads to our second point.

It’s interesting that Hopf speaks of “exact natural science.”  Today much of science and engineering is driven by probabalistic considerations, which in turn lead to statistical analysis.  Probability and statistics is a very useful tool, but not a substitute for the understanding of the actual mechanisms by which things work.  The actual mechanisms (physical laws, etc.) are what cause the phenomena which we record as statistics, not the other way around.  The fact that there are variations in these should not blind us to the core reality.

The advent of computers with broad-based number crunching abilities has only inflated our overconfidence in such methods.  It is essential, however, that we understand the why of phenomena as well as the what.  We must both be able to quantify the results and the correct causes of what is going on around us.  Two recent debacles illustrate this.

The first is the climate change fiasco we’ve been treated to of late.  Removing the dissimulation (as opposed to simulation) of some involved in the science, the core problem is that we do not as of yet have a model of global climate sufficiently comprehensive so that we can dispense with reliance on the statistics and project what will happen with a reasonable degree of confidence.  Part of the problem is the core problem in chaos theory: minor variations in initial conditions lead to major variations in the results.  But without such a model we are bereft with a definitive “why” as much as “what.”

The second is our financial collapse.  The models developed of the elaborate credit structure were fine as far as they went.  But ultimately they were divorced from sustainable reality because they did not take in to consideration all of the factors, many of which were obvious to those with raw experience.

The issue of causality is one that is central to our understanding of the universe.

The equations of motion for linear vibrating systems are well known and widely used in both mechanical and electrical devices. However, when students are introduced to these, they are frequently presented with solutions which are either essentially underived or inadequately so.

This brief presentation will attempt to address this deficiency and hopefully show the derivation of the equation of motion for an undamped oscillating system in a more rigourous way.

Consider a simple spring/mass system without a forcing function. The equation of motion can be expressed as

where x(t) is displacement as a function of time, m is the mass of the system, and k is the spring constant. The negative sign on the right hand side of the equation is not an accident, as the spring force always opposes the motion of the mass, and is the result of using a mechanical engineers’ “free body diagram” method to develop the equation.

Solutions to this equation generally run in two forms. The first is a sum of sines and cosines:

But it’s more common to see it in the form of

The latter is simpler and easier to apply; however, it is seldom derived as much as assumed. So how can it be obtained from the original equation?

Let us begin by considering the original differential equation. With its constant coefficients, the most straightforward solution would be a solution where the derivative (and we, of course, would derive it twice) would be itself. This is the case where the function is exponential, so let us assume the equation to be in the form of

(I had an interesting fluid mechanics/heat transfer teacher who would say about this step that “you just write the answer down,” which we as his students found exasperating, but this method minimises that.)

Substituting this into the original equation of motion and diving out the identical exponentials yields

Solving for α yields

The right hand term is the natural frequency of the system, more generally expressed as a real number:

Thus for simplicity the solution can be written as

At this point it is not clear which of these two solutions is correct, so let us write the general solution as

Because of the complex exponential definition of sines and cosines, we see the beginning of a solution in simply one or the other, but at this point the coefficients are in the way.

These coefficients are determined from the initial conditions. Let us consider these at t=0:

Substituting these into our assumed general solution yields

The coefficients then solve to

It is noteworthy that the two coefficients are complex conjugates of each other.

Since the general solution is written in exponential form, it makes sense that, if the coefficients are to be removed so we can enable a direct solution to a sine or cosine, they too should be in exponential form. Converting the two coefficients to polar form yields

Substituting these coefficients into the general solution, we have

Factoring out the radical and recognising that the arctangent is an odd function,

The quantity in brackets is the complex exponential definition of the cosine, since the two exponents are negatives of each other. The solution can thus be written as

If we define

the solution is

which can be rewritten in a number of ways.

If the dampening is added, the problem can be solved in the same way, but the algebra is a little more complicated, and we will end up additionally with a real exponential (decay) in the final solution.

This derivation demonstrates the power of complex analysis as applied to differential equations even in a simple way.

More examples of this kind of thing are here and here.